In this document, we intend to investigate the following key questions, assuming a fixed array of \(10^6\) SNPs and a quantitative trait in which SNP effect sizes follow a normal distribution:
In this section, we look at the average number of significant SNPs, the average proportion of these significant SNPs that have association estimates more extreme than their true effect size and the average MSE of significant SNPs at two different thresholds; the common genome-wide significance threshold of \(5 \times 10^{-8}\) and a higher threshold of \(5 \times 10^{-4}\). We consider these properties under certain combinations of values for the following parameters:
n_samplesh2prop_effectSThe 24 different combinations that we will investigate throughout this document are detailed below:
| Scenario | n_samples | h2 | prop_effect | S |
|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 |
| 2 | 300,000 | 0.3 | 0.010 | -1 |
| 3 | 30,000 | 0.8 | 0.010 | -1 |
| 4 | 300,000 | 0.8 | 0.010 | -1 |
| 5 | 30,000 | 0.3 | 0.001 | -1 |
| 6 | 300,000 | 0.3 | 0.001 | -1 |
| 7 | 30,000 | 0.8 | 0.001 | -1 |
| 8 | 300,000 | 0.8 | 0.001 | -1 |
| 9 | 30,000 | 0.3 | 0.010 | 0 |
| 10 | 300,000 | 0.3 | 0.010 | 0 |
| 11 | 30,000 | 0.8 | 0.010 | 0 |
| 12 | 300,000 | 0.8 | 0.010 | 0 |
| Scenario | n_samples | h2 | prop_effect | S |
|---|---|---|---|---|
| 13 | 30,000 | 0.3 | 0.001 | 0 |
| 14 | 300,000 | 0.3 | 0.001 | 0 |
| 15 | 30,000 | 0.8 | 0.001 | 0 |
| 16 | 300,000 | 0.8 | 0.001 | 0 |
| 17 | 30,000 | 0.3 | 0.010 | 1 |
| 18 | 300,000 | 0.3 | 0.010 | 1 |
| 19 | 30,000 | 0.8 | 0.010 | 1 |
| 20 | 300,000 | 0.8 | 0.010 | 1 |
| 21 | 30,000 | 0.3 | 0.001 | 1 |
| 22 | 300,000 | 0.3 | 0.001 | 1 |
| 23 | 30,000 | 0.8 | 0.001 | 1 |
| 24 | 300,000 | 0.8 | 0.001 | 1 |
\(~\) \(~\) \(~\)
Running the code provided in nsig_prop_bias_100sim.R, we obtain the following results:
| Scenario | n_samples | h2 | prop_effect | S | n_sig 5e-8 | prop_bias 5e-8 | mse 5e-8 | n_sig 5e-4 | prop_bias 5e-4 | mse 5e-4 | sd(n_sig) 5e-8 | sd(prop_bias) 5e-8 | sd(mse) 5e-8 | sd(n_sig) 5e-4 | sd(prop_bias) 5e-4 | sd(mse) 5e-4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 0.70 | 1.0000 | 0.001573 | 612.78 | 0.9997 | 0.001900 | 0.745 | 0.0000 | 0.001234 | 24.841 | 0.0007 | 0.000108 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | 848.63 | 0.7619 | 0.000022 | 3200.03 | 0.7473 | 0.000049 | 18.145 | 0.0142 | 0.000002 | 38.450 | 0.0075 | 0.000002 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | 31.85 | 0.9804 | 0.000598 | 1083.65 | 0.9591 | 0.001186 | 5.208 | 0.0280 | 0.000198 | 32.603 | 0.0054 | 0.000063 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | 2760.68 | 0.6284 | 0.000017 | 5362.80 | 0.6391 | 0.000035 | 30.091 | 0.0084 | 0.000001 | 52.753 | 0.0066 | 0.000001 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | 86.90 | 0.7591 | 0.000214 | 774.49 | 0.8946 | 0.001480 | 6.317 | 0.0437 | 0.000054 | 22.626 | 0.0097 | 0.000078 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 568.70 | 0.5509 | 0.000016 | 1215.01 | 0.7302 | 0.000099 | 14.074 | 0.0225 | 0.000002 | 29.951 | 0.0139 | 0.000006 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | 276.26 | 0.6273 | 0.000168 | 987.38 | 0.8037 | 0.001183 | 10.413 | 0.0271 | 0.000027 | 25.221 | 0.0124 | 0.000066 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 727.10 | 0.5257 | 0.000016 | 1324.22 | 0.7036 | 0.000092 | 12.491 | 0.0171 | 0.000001 | 25.793 | 0.0120 | 0.000006 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | 1.45 | 1.0000 | 0.001661 | 622.50 | 0.9980 | 0.001822 | 1.298 | 0.0000 | 0.005559 | 25.639 | 0.0018 | 0.000089 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | 882.45 | 0.7297 | 0.000012 | 3059.66 | 0.7442 | 0.000046 | 18.435 | 0.0152 | 0.000001 | 41.677 | 0.0064 | 0.000003 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | 48.06 | 0.9509 | 0.000245 | 1115.40 | 0.9402 | 0.001088 | 6.350 | 0.0301 | 0.000050 | 30.487 | 0.0070 | 0.000065 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | 2586.78 | 0.6204 | 0.000011 | 5034.98 | 0.6449 | 0.000032 | 32.771 | 0.0096 | 0.000000 | 50.379 | 0.0062 | 0.000002 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | 88.32 | 0.7254 | 0.000116 | 752.41 | 0.8953 | 0.001489 | 6.377 | 0.0480 | 0.000025 | 23.212 | 0.0104 | 0.000070 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 531.27 | 0.5560 | 0.000012 | 1182.74 | 0.7392 | 0.000100 | 12.431 | 0.0215 | 0.000001 | 26.240 | 0.0107 | 0.000006 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 257.44 | 0.6234 | 0.000106 | 946.54 | 0.8109 | 0.001207 | 8.936 | 0.0276 | 0.000013 | 23.704 | 0.0121 | 0.000058 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 691.46 | 0.5268 | 0.000013 | 1297.90 | 0.7097 | 0.000093 | 13.526 | 0.0185 | 0.000001 | 24.238 | 0.0118 | 0.000005 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | 2.55 | 0.9941 | 0.000610 | 639.55 | 0.9959 | 0.001760 | 1.623 | 0.0405 | 0.000674 | 26.371 | 0.0024 | 0.000087 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | 919.28 | 0.7031 | 0.000010 | 2905.76 | 0.7332 | 0.000046 | 15.799 | 0.0142 | 0.000001 | 37.008 | 0.0070 | 0.000003 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | 68.13 | 0.9178 | 0.000198 | 1148.37 | 0.9199 | 0.001029 | 7.795 | 0.0370 | 0.000087 | 30.266 | 0.0080 | 0.000056 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | 2433.27 | 0.6105 | 0.000009 | 4634.00 | 0.6436 | 0.000033 | 30.705 | 0.0107 | 0.000000 | 45.304 | 0.0058 | 0.000001 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | 93.15 | 0.7056 | 0.000096 | 741.13 | 0.8962 | 0.001513 | 5.960 | 0.0428 | 0.000015 | 27.299 | 0.0101 | 0.000095 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 482.94 | 0.5516 | 0.000009 | 1122.87 | 0.7510 | 0.000104 | 12.742 | 0.0244 | 0.000001 | 26.147 | 0.0100 | 0.000006 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 244.29 | 0.6120 | 0.000089 | 916.49 | 0.8199 | 0.001243 | 9.237 | 0.0272 | 0.000010 | 26.753 | 0.0122 | 0.000074 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 632.74 | 0.5336 | 0.000010 | 1241.42 | 0.7219 | 0.000095 | 14.730 | 0.0194 | 0.000001 | 26.609 | 0.0125 | 0.000005 |
\(~\) \(~\) \(~\)
It is important to note here that for scenarios 1, 9 and 17, very few significant SNPs are detected on average. In some instances, we may even find that no SNPs are deemed significant at a threshold of \(5 \times 10^{-8}\). We must keep this observation in mind going forward as we investigate the performance of methods under these three scenarios.
For both thresholds, the average number of significant SNPs increases as sample size increases, as expected. It also increases with heritability. However, the effect of changing prop_effect is more interesting. Decreasing the proportion of effect SNPs from 0.01 to 0.001 results in the number of significant SNPs increasing for a sample size of 30,000 while we witness the number of SNPs passing the genome-wide significance threshold decreasing for a larger sample size of 300,000.
Furthermore, increasing sample size and increasing heritability from 0.3 to 0.8 all tend to decrease the fraction of significant SNPs whose estimates are more extreme than their true effect size. Decreasing polygenicity from 0.01 to 0.001 also has this same effect at a significance threshold of \(5 \times 10^{-8}\).
In order to gain a better insight into the information detailed in the above table, we simulate a single set of GWAS summary statistics and plot \(z\) vs \(\text{bias}\) in which \(\text{bias} = \hat\beta - \beta\) for each of the 24 different scenarios. On all figures, the bright red line corresponds to the significance threshold of \(5 \times 10^{-8}\) while the darker red line relates to \(5 \times 10^{-4}\).
Using the code detailed in norm_5e-8_10sim.R and a total of 10 simulations, we evaluated six different Winner’s Curse methods across each of the 24 scenarios using the following three bias evaluation metrics:
flbmserel_mseNote: All averages are obtained over only those simulations in which at least one significant SNP was detected.
Firstly, the fraction of \(n\) significant SNPs in which their association estimates are less biased due to method implementation may be mathematically described as: \[\frac{1}{n} \; \sum_{i=1}^{n}\mathbb{I} \left\{ \left| \hat\beta_i - \beta_i \right| > \left|\hat\beta_{\text{adj,}i} - \beta_i\right| \right\},\]in which \(\left| \frac{\hat\beta_i}{\hat\sigma_i} \right| > Z_{\frac{\alpha}{2}}\) for all \(i = 1,...,n\), where \(\hat\beta_i\) is the estimated naive effect size of SNP \(i\), \(\beta_i\) is its true effect size and \(\hat\beta_{\text{adj,}i}\) is its new effect size estimate obtained as a result of application of the Winner’s Curse adjustment method of interest. The significance threshold is represented by \(\alpha\).
Using the same notation, the average MSE over \(n\) significant SNPs is defined as: \[\frac{1}{n} \sum^n_{i=1} (\hat\beta_i - \beta_i)^2.\] Thus, using the above, we may formally define the change in average MSE of significant SNPs as: \[\frac{1}{n} \sum^n_{i=1} (\hat\beta_{\text{adj,}i} - \beta_i)^2 - \frac{1}{n} \sum^n_{i=1} (\hat\beta_i - \beta_i)^2\] and the relative change in average MSE of significant SNPs as: \[\frac{\frac{1}{n} \sum^n_{i=1} (\hat\beta_{\text{adj,}i} - \beta_i)^2 - \frac{1}{n} \sum^n_{i=1} (\hat\beta_i - \beta_i)^2}{\frac{1}{n} \sum^n_{i=1} (\hat\beta_i - \beta_i)^2}.\]
\(~\)
Results of the simulations are plotted. Error bars are also included in the plots. These figures allow us to see more clearly the scenarios in which it would be beneficial to apply a Winner’s Curse correction method and also, provide us with a better indication of which method we should use.
Summary of results for flb contained in norm_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 1.0000 | 1.0000 | 1.0000 | 0.9444 | 1.0000 | 1.0000 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | 0.6465 | 0.4422 | 0.6078 | 0.5290 | 0.5049 | 0.5138 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | 0.8838 | 0.7446 | 0.7949 | 0.6342 | 0.7626 | 0.6896 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | 0.5681 | 0.2803 | 0.5299 | 0.5149 | 0.4874 | 0.4986 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | 0.6840 | 0.3621 | 0.5168 | 0.5419 | 0.5012 | 0.4933 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.4882 | 0.1447 | 0.2824 | 0.5031 | 0.4741 | 0.4943 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | 0.5195 | 0.2095 | 0.3149 | 0.5085 | 0.4758 | 0.4878 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.4703 | 0.1016 | 0.2803 | 0.5137 | 0.4648 | 0.4934 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | 0.8889 | 1.0000 | 0.9238 | 0.9333 | 0.8889 | 0.9167 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | 0.6178 | 0.4004 | 0.5783 | 0.5233 | 0.4975 | 0.5189 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | 0.8420 | 0.6658 | 0.7481 | 0.6007 | 0.6478 | 0.5947 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | 0.5568 | 0.2688 | 0.5268 | 0.5125 | 0.4857 | 0.4969 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | 0.6267 | 0.3232 | 0.4546 | 0.5343 | 0.4828 | 0.4807 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.4802 | 0.1446 | 0.2737 | 0.5164 | 0.4768 | 0.5002 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 0.5025 | 0.2091 | 0.2759 | 0.5143 | 0.4810 | 0.5001 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.4821 | 0.1072 | 0.2917 | 0.4994 | 0.4693 | 0.4940 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | 0.8021 | 0.8883 | 0.8020 | 0.7852 | 0.8611 | 0.7370 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | 0.6089 | 0.3625 | 0.5485 | 0.5336 | 0.4955 | 0.5182 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | 0.8150 | 0.5901 | 0.6627 | 0.5702 | 0.6108 | 0.5913 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | 0.5501 | 0.2605 | 0.5135 | 0.5152 | 0.4860 | 0.5015 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | 0.5419 | 0.2678 | 0.4008 | 0.5413 | 0.4756 | 0.5264 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.4867 | 0.1301 | 0.2748 | 0.5046 | 0.4663 | 0.4881 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.4988 | 0.1816 | 0.2746 | 0.4995 | 0.4670 | 0.5049 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.4588 | 0.1123 | 0.2948 | 0.4999 | 0.4688 | 0.4755 |
\(~\) \(~\) \(~\) \(~\)
Fraction of significant SNPs less biased due to method implementation, using a significance threshold of \(5 \times 10^{-8}\):
\(~\) \(~\) \(~\) \(~\)
Summary of results for mse contained in norm_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | -0.001001 | -0.003130 | -0.001256 | -0.000626 | -0.001466 | -0.001470 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.000007 | 0.000000 | -0.000005 | 0.000055 | 0.000022 | 0.000034 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.000433 | -0.000434 | -0.000483 | 0.000273 | -0.000324 | 0.000008 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.000002 | 0.000005 | 0.000000 | 0.000033 | 0.000021 | 0.000026 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.000071 | 0.000180 | 0.000023 | 0.000526 | 0.000261 | 0.000438 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.000000 | 0.000006 | 0.000013 | 0.000017 | 0.000009 | 0.000012 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.000005 | 0.000118 | 0.000164 | 0.000375 | 0.000202 | 0.000265 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.000001 | 0.000004 | 0.000007 | 0.000008 | 0.000022 | 0.000012 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.000837 | -0.000438 | -0.000583 | -0.000180 | -0.000545 | -0.000580 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.000003 | 0.000001 | -0.000002 | 0.000027 | 0.000011 | 0.000018 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.000148 | -0.000145 | -0.000167 | 0.000156 | -0.000057 | 0.000055 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.000001 | 0.000002 | -0.000001 | 0.000021 | 0.000011 | 0.000015 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.000026 | 0.000061 | 0.000024 | 0.000243 | 0.000129 | 0.000185 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.000000 | 0.000005 | 0.000009 | 0.000014 | 0.000009 | 0.000009 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | -0.000001 | 0.000061 | 0.000096 | 0.000193 | 0.000121 | 0.000165 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.000001 | 0.000004 | 0.000006 | 0.000011 | 0.000022 | 0.000014 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.000305 | -0.000368 | -0.000368 | -0.000215 | -0.000548 | -0.000187 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.000002 | 0.000002 | -0.000001 | 0.000023 | 0.000010 | 0.000015 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.000116 | -0.000084 | -0.000100 | 0.000223 | -0.000016 | 0.000081 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.000001 | 0.000002 | 0.000000 | 0.000016 | 0.000009 | 0.000012 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.000007 | 0.000079 | 0.000053 | 0.000229 | 0.000118 | 0.000131 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.000000 | 0.000004 | 0.000008 | 0.000012 | 0.000007 | 0.000008 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.000003 | 0.000063 | 0.000102 | 0.000173 | 0.000097 | 0.000117 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.000001 | 0.000002 | 0.000005 | 0.000009 | 0.000037 | 0.000018 |
\(~\) \(~\) \(~\) \(~\)
Change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-8}\):
\(~\) \(~\) \(~\) \(~\)
Summary of results for rel_mse contained in norm_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 0.0507 | -0.9492 | -0.9003 | -0.6770 | -0.8856 | -0.8814 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.3318 | 0.0221 | -0.2262 | 2.6268 | 1.0571 | 1.6423 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.7151 | -0.6603 | -0.7701 | 0.5325 | -0.5115 | 0.0245 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.0962 | 0.3036 | -0.0255 | 1.9721 | 1.2329 | 1.5102 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.2889 | 0.9262 | 0.1466 | 2.3456 | 1.1771 | 2.3775 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.0258 | 0.4226 | 0.8421 | 1.0807 | 0.5761 | 0.7620 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.0257 | 0.7143 | 1.0937 | 2.1556 | 1.3581 | 1.4677 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.0645 | 0.2524 | 0.4507 | 0.5107 | 1.4334 | 0.7650 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.8664 | -0.8423 | -0.7283 | -0.1695 | -0.8529 | -0.6569 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.2659 | 0.0530 | -0.1682 | 2.3019 | 0.9933 | 1.5287 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.6536 | -0.5714 | -0.6742 | 0.7167 | -0.2226 | 0.2667 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.0971 | 0.2097 | -0.0475 | 1.9913 | 1.0455 | 1.3835 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.2386 | 0.5800 | 0.2616 | 2.2566 | 1.1942 | 1.8102 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.0410 | 0.4408 | 0.8377 | 1.1782 | 0.8077 | 0.8149 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | -0.0116 | 0.6069 | 0.9116 | 1.8770 | 1.1874 | 1.5703 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.0698 | 0.3047 | 0.4743 | 0.8816 | 1.7814 | 1.1572 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.2448 | -0.8222 | -0.3110 | 0.1454 | -0.7682 | -0.2974 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.2176 | 0.1789 | -0.0560 | 2.3831 | 1.0067 | 1.5483 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.6034 | -0.4314 | -0.5516 | 1.1932 | -0.0643 | 0.4635 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.0754 | 0.2280 | 0.0061 | 1.8173 | 1.0108 | 1.3115 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.0737 | 0.8246 | 0.5939 | 2.3955 | 1.2562 | 1.2410 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.0448 | 0.4270 | 0.8541 | 1.2463 | 0.7071 | 0.8717 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.0351 | 0.7122 | 1.1913 | 1.8746 | 1.1730 | 1.3601 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.1411 | 0.2342 | 0.5176 | 0.8523 | 3.6076 | 1.7753 |
\(~\) \(~\) \(~\) \(~\)
Relative change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-8}\):
Similar to part 2 above, we use the code detailed in norm_5e-4_10sim.R with a total of 10 simulations in order to evaluate six different Winner’s Curse methods across each of the 24 scenarios. The same three bias evaluation metrics are considered.
Summary of results for flb contained in norm_5e-4_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 0.9454 | 0.9449 | 0.9724 | 0.9695 | 0.9908 | 0.9810 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | 0.5998 | 0.4608 | 0.5225 | 0.6079 | 0.6104 | 0.6041 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | 0.7565 | 0.7121 | 0.7531 | 0.7961 | 0.8548 | 0.8243 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | 0.5487 | 0.3505 | 0.4950 | 0.5632 | 0.5541 | 0.5527 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | 0.7917 | 0.7348 | 0.7656 | 0.8377 | 0.8404 | 0.8363 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.6661 | 0.4972 | 0.5711 | 0.7068 | 0.6953 | 0.7118 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | 0.7078 | 0.6059 | 0.6280 | 0.7589 | 0.7546 | 0.7597 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.6461 | 0.4438 | 0.5529 | 0.6983 | 0.6785 | 0.6876 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | 0.9257 | 0.9170 | 0.9541 | 0.9556 | 0.9836 | 0.9656 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | 0.5921 | 0.4611 | 0.5249 | 0.6127 | 0.6076 | 0.6077 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | 0.7472 | 0.6949 | 0.7264 | 0.7870 | 0.8369 | 0.8084 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | 0.5487 | 0.3547 | 0.5040 | 0.5672 | 0.5561 | 0.5614 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | 0.7988 | 0.7340 | 0.7672 | 0.8391 | 0.8475 | 0.8430 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.6666 | 0.5070 | 0.5683 | 0.7161 | 0.6998 | 0.7102 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 0.7119 | 0.6147 | 0.6375 | 0.7682 | 0.7640 | 0.7697 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.6521 | 0.4562 | 0.5575 | 0.6935 | 0.6720 | 0.6790 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | 0.8997 | 0.8978 | 0.9285 | 0.9448 | 0.9684 | 0.9548 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | 0.5944 | 0.4521 | 0.5157 | 0.6140 | 0.6135 | 0.6090 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | 0.7288 | 0.6709 | 0.7072 | 0.7641 | 0.8077 | 0.7797 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | 0.5524 | 0.3568 | 0.5034 | 0.5718 | 0.5594 | 0.5648 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | 0.7987 | 0.7440 | 0.7681 | 0.8492 | 0.8467 | 0.8481 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.6791 | 0.5203 | 0.5893 | 0.7270 | 0.7234 | 0.7250 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.7313 | 0.6308 | 0.6617 | 0.7742 | 0.7757 | 0.7785 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.6465 | 0.4756 | 0.5691 | 0.7071 | 0.6843 | 0.7013 |
\(~\) \(~\) \(~\) \(~\)
Fraction of significant SNPs less biased due to method implementation, using a significance threshold of \(5 \times 10^{-4}\):
\(~\) \(~\) \(~\) \(~\)
Summary of results for mse contained in norm_5e-4_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | -0.001804 | -0.001801 | -0.001738 | -0.001627 | -0.001359 | -0.001496 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.000020 | -0.000011 | -0.000015 | -0.000012 | -0.000018 | -0.000016 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.000897 | -0.000889 | -0.000896 | -0.000799 | -0.000779 | -0.000826 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.000010 | -0.000001 | -0.000006 | -0.000004 | -0.000007 | -0.000007 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.001187 | -0.001180 | -0.001078 | -0.001162 | -0.000976 | -0.001079 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | -0.000077 | -0.000069 | -0.000058 | -0.000071 | -0.000062 | -0.000068 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.000895 | -0.000824 | -0.000727 | -0.000876 | -0.000762 | -0.000846 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | -0.000066 | -0.000065 | -0.000054 | -0.000068 | -0.000047 | -0.000060 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.001741 | -0.001753 | -0.001602 | -0.001492 | -0.001276 | -0.001402 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.000023 | -0.000022 | -0.000021 | -0.000022 | -0.000023 | -0.000023 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.000849 | -0.000875 | -0.000815 | -0.000802 | -0.000736 | -0.000834 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.000013 | -0.000008 | -0.000010 | -0.000009 | -0.000012 | -0.000011 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.001291 | -0.001264 | -0.001195 | -0.001171 | -0.001000 | -0.001160 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | -0.000078 | -0.000070 | -0.000057 | -0.000072 | -0.000065 | -0.000072 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | -0.000979 | -0.000940 | -0.000851 | -0.000931 | -0.000786 | -0.000888 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | -0.000065 | -0.000065 | -0.000054 | -0.000067 | -0.000043 | -0.000059 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.001656 | -0.001708 | -0.001548 | -0.001517 | -0.001213 | -0.001378 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.000024 | -0.000023 | -0.000022 | -0.000024 | -0.000024 | -0.000026 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.000828 | -0.000830 | -0.000795 | -0.000802 | -0.000677 | -0.000753 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.000014 | -0.000012 | -0.000012 | -0.000013 | -0.000014 | -0.000015 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.001351 | -0.001330 | -0.001184 | -0.001192 | -0.000998 | -0.001148 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | -0.000083 | -0.000076 | -0.000064 | -0.000081 | -0.000067 | -0.000076 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | -0.001087 | -0.001002 | -0.000893 | -0.001002 | -0.000821 | -0.000902 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | -0.000072 | -0.000071 | -0.000057 | -0.000073 | -0.000041 | -0.000065 |
\(~\) \(~\) \(~\) \(~\)
Change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-4}\):
\(~\) \(~\) \(~\) \(~\)
Summary of results for rel_mse contained in norm_5e-4_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | -0.9495 | -0.9531 | -0.8988 | -0.8988 | -0.8988 | -0.8988 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.4039 | -0.2233 | -0.3032 | -0.3032 | -0.3032 | -0.3032 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.7519 | -0.7415 | -0.7406 | -0.7406 | -0.7406 | -0.7406 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.2785 | -0.0289 | -0.1681 | -0.1681 | -0.1681 | -0.1681 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.7974 | -0.7718 | -0.7368 | -0.7368 | -0.7368 | -0.7368 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | -0.7650 | -0.6881 | -0.5645 | -0.5645 | -0.5645 | -0.5645 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.7600 | -0.7027 | -0.6101 | -0.6101 | -0.6101 | -0.6101 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | -0.7266 | -0.6984 | -0.5693 | -0.5693 | -0.5693 | -0.5693 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.9581 | -0.9610 | -0.8900 | -0.8900 | -0.8900 | -0.8900 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.5074 | -0.4572 | -0.4590 | -0.4590 | -0.4590 | -0.4590 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.7960 | -0.8027 | -0.7645 | -0.7645 | -0.7645 | -0.7645 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.3798 | -0.2440 | -0.3148 | -0.3148 | -0.3148 | -0.3148 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.8686 | -0.8498 | -0.7890 | -0.7890 | -0.7890 | -0.7890 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | -0.7685 | -0.7035 | -0.5816 | -0.5816 | -0.5816 | -0.5816 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | -0.8155 | -0.7747 | -0.6954 | -0.6954 | -0.6954 | -0.6954 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | -0.7139 | -0.6916 | -0.5784 | -0.5784 | -0.5784 | -0.5784 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.9419 | -0.9543 | -0.8820 | -0.8820 | -0.8820 | -0.8820 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.5347 | -0.4905 | -0.4819 | -0.4819 | -0.4819 | -0.4819 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.7932 | -0.7945 | -0.7530 | -0.7530 | -0.7530 | -0.7530 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.4397 | -0.3487 | -0.3736 | -0.3736 | -0.3736 | -0.3736 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.8957 | -0.8726 | -0.7953 | -0.7953 | -0.7953 | -0.7953 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | -0.8130 | -0.7587 | -0.6277 | -0.6277 | -0.6277 | -0.6277 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | -0.8561 | -0.8119 | -0.7287 | -0.7287 | -0.7287 | -0.7287 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | -0.7568 | -0.7387 | -0.6040 | -0.6040 | -0.6040 | -0.6040 |
\(~\) \(~\) \(~\) \(~\)
Relative change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-4}\):
Here we investigate the 24 different scenarios under a skewed distribution of effect sizes. In order to create a bimodal distribution, we simulate 50% of effect sizes of the true effect SNPs from a normal distribution centered at 0 while the other half are generated from a normal distribution with mean 2.5. As above, we first have a look at the expected number of significant SNPs and the expected proportion of those in which their association estimate is exaggerated.
Running the code provided in nsig_prop_bias_100sim.R, we obtain the following results:
| Scenario | n_samples | h2 | prop_effect | S | n_sig 5e-8 | prop_bias 5e-8 | mse 5e-8 | sd(n_sig) 5e-8 | sd(prop_bias) 5e-8 | sd(mse) 5e-8 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 0.49 | 1.0000 | 0.001239 | 0.659 | 0.0000 | 0.000966 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | 857.47 | 0.7724 | 0.000016 | 21.728 | 0.0140 | 0.000002 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | 24.81 | 0.9966 | 0.000442 | 4.890 | 0.0126 | 0.000117 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | 2821.07 | 0.6250 | 0.000014 | 29.028 | 0.0085 | 0.000001 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | 86.72 | 0.7677 | 0.000156 | 6.909 | 0.0434 | 0.000037 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 568.01 | 0.5494 | 0.000014 | 12.738 | 0.0178 | 0.000002 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | 281.94 | 0.6214 | 0.000138 | 9.623 | 0.0309 | 0.000024 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 729.36 | 0.5221 | 0.000015 | 11.343 | 0.0179 | 0.000001 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | 0.54 | 1.0000 | 0.001643 | 0.731 | 0.0000 | 0.003033 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | 911.67 | 0.7746 | 0.000012 | 22.312 | 0.0129 | 0.000001 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | 23.16 | 0.9987 | 0.000392 | 4.683 | 0.0079 | 0.000069 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | 2859.69 | 0.6116 | 0.000010 | 29.929 | 0.0097 | 0.000000 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | 90.54 | 0.7672 | 0.000113 | 6.641 | 0.0470 | 0.000020 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 541.99 | 0.5429 | 0.000012 | 12.748 | 0.0191 | 0.000001 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 285.65 | 0.6135 | 0.000104 | 9.134 | 0.0300 | 0.000014 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 690.16 | 0.5257 | 0.000013 | 11.850 | 0.0204 | 0.000001 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | 0.47 | 1.0000 | 0.001354 | 0.745 | 0.0000 | 0.000857 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | 917.61 | 0.8198 | 0.000012 | 23.791 | 0.0108 | 0.000001 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | 15.15 | 1.0000 | 0.000484 | 2.949 | 0.0000 | 0.000087 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | 3134.20 | 0.6022 | 0.000010 | 25.334 | 0.0093 | 0.000000 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | 91.80 | 0.8184 | 0.000123 | 6.123 | 0.0373 | 0.000022 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 522.21 | 0.5321 | 0.000012 | 8.395 | 0.0192 | 0.000001 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 313.70 | 0.6046 | 0.000096 | 9.802 | 0.0235 | 0.000010 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 628.33 | 0.5248 | 0.000014 | 10.876 | 0.0186 | 0.000001 |
\(~\) \(~\) \(~\) \(~\)
Next, we repeat the process illustrated in Section 2 using the same bias evaluation metrics with a significance threshold of \(5 \times 10^{-8}\).
Summary of results for flb contained in skew_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | 0.6602 | 0.4526 | 0.6297 | 0.5353 | 0.5017 | 0.5087 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | 0.9320 | 0.8476 | 0.8899 | 0.6292 | 0.8174 | 0.7111 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | 0.5632 | 0.2800 | 0.5431 | 0.5125 | 0.4865 | 0.4974 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | 0.6102 | 0.3468 | 0.5279 | 0.5340 | 0.5144 | 0.5307 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.4831 | 0.1459 | 0.2915 | 0.5022 | 0.4736 | 0.4830 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | 0.5371 | 0.2252 | 0.3257 | 0.5174 | 0.4789 | 0.4968 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.4698 | 0.0984 | 0.2784 | 0.4953 | 0.4788 | 0.4828 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | 0.6457 | 0.4562 | 0.6293 | 0.5325 | 0.4980 | 0.5059 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | 0.9630 | 0.8636 | 0.9338 | 0.6971 | 0.8777 | 0.6586 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | 0.5554 | 0.2651 | 0.5347 | 0.5090 | 0.4864 | 0.5031 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | 0.6808 | 0.3087 | 0.5335 | 0.4939 | 0.4977 | 0.5004 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.4798 | 0.1316 | 0.2796 | 0.4985 | 0.4685 | 0.5068 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 0.5019 | 0.2051 | 0.2981 | 0.5115 | 0.4946 | 0.4934 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.4842 | 0.1064 | 0.2774 | 0.4993 | 0.4725 | 0.5010 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | 0.6860 | 0.5203 | 0.6782 | 0.5451 | 0.4867 | 0.5060 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | 0.9923 | 0.9563 | 0.9778 | 0.6972 | 0.8984 | 0.8678 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | 0.5422 | 0.2553 | 0.5372 | 0.5129 | 0.4807 | 0.4942 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | 0.6793 | 0.3596 | 0.6099 | 0.5534 | 0.4918 | 0.5101 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.4772 | 0.1125 | 0.2491 | 0.4955 | 0.4535 | 0.4924 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.5083 | 0.1797 | 0.2977 | 0.5154 | 0.4805 | 0.5028 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.4787 | 0.0890 | 0.2545 | 0.5117 | 0.4779 | 0.4917 |
\(~\) \(~\) \(~\) \(~\)
Fraction of significant SNPs less biased due to method implementation, using a significance threshold of \(5 \times 10^{-8}\) and a skewed distribution of effect sizes:
\(~\) \(~\) \(~\) \(~\)
Summary of results for mse contained in skew_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | -0.001274 | -0.001701 | -0.000495 | -0.000563 | -0.000640 | -0.001265 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.000007 | -0.000002 | -0.000006 | 0.000038 | 0.000015 | 0.000024 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.000389 | -0.000346 | -0.000388 | 0.000060 | -0.000262 | -0.000136 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.000002 | 0.000003 | -0.000001 | 0.000030 | 0.000016 | 0.000022 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.000035 | 0.000067 | -0.000004 | 0.000434 | 0.000117 | 0.000272 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.000001 | 0.000007 | 0.000012 | 0.000015 | 0.000011 | 0.000014 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.000010 | 0.000072 | 0.000088 | 0.000369 | 0.000185 | 0.000214 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.000001 | 0.000004 | 0.000009 | 0.000010 | 0.000011 | 0.000007 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.001155 | -0.000693 | -0.001221 | -0.000859 | -0.000830 | -0.000999 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.000004 | -0.000001 | -0.000004 | 0.000030 | 0.000012 | 0.000018 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.000294 | -0.000356 | -0.000338 | -0.000019 | -0.000291 | -0.000088 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.000001 | 0.000003 | -0.000001 | 0.000021 | 0.000012 | 0.000015 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.000039 | 0.000063 | -0.000014 | 0.000354 | 0.000102 | 0.000190 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.000000 | 0.000004 | 0.000006 | 0.000013 | 0.000008 | 0.000011 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 0.000001 | 0.000062 | 0.000053 | 0.000206 | 0.000107 | 0.000154 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.000001 | 0.000004 | 0.000007 | 0.000012 | 0.000009 | 0.000009 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.001416 | -0.000878 | -0.001417 | -0.001012 | -0.000918 | -0.001379 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.000006 | -0.000003 | -0.000006 | 0.000029 | 0.000011 | 0.000019 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.000390 | -0.000419 | -0.000449 | -0.000168 | -0.000368 | -0.000272 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.000001 | 0.000003 | -0.000001 | 0.000021 | 0.000012 | 0.000016 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.000046 | 0.000039 | -0.000038 | 0.000272 | 0.000128 | 0.000198 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.000001 | 0.000004 | 0.000007 | 0.000014 | 0.000008 | 0.000010 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.000002 | 0.000075 | 0.000044 | 0.000203 | 0.000118 | 0.000139 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.000001 | 0.000004 | 0.000007 | 0.000009 | 0.000009 | 0.000006 |
\(~\) \(~\) \(~\) \(~\)
Change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-8}\) and a skewed distribution of effect sizes:
\(~\) \(~\) \(~\) \(~\)
Summary of results for rel_mse contained in skew_5e-8_10sim.csv:
| Scenario | n_samples | h2 | prop_effect | S | EB | FIQT | BR | cl1 | cl2 | cl3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 30,000 | 0.3 | 0.010 | -1 | -0.8635 | -0.9212 | -0.8219 | -0.4293 | -0.8107 | -0.8705 |
| 2 | 300,000 | 0.3 | 0.010 | -1 | -0.4172 | -0.1148 | -0.3705 | 2.3193 | 0.8922 | 1.5814 |
| 3 | 30,000 | 0.8 | 0.010 | -1 | -0.8060 | -0.8335 | -0.8544 | 0.1729 | -0.6060 | -0.2871 |
| 4 | 300,000 | 0.8 | 0.010 | -1 | -0.1246 | 0.2416 | -0.0750 | 2.2128 | 1.1484 | 1.6053 |
| 5 | 30,000 | 0.3 | 0.001 | -1 | -0.1651 | 0.5245 | 0.0929 | 3.1574 | 0.7103 | 1.4561 |
| 6 | 300,000 | 0.3 | 0.001 | -1 | 0.0382 | 0.4855 | 0.8315 | 1.0604 | 0.7592 | 1.0271 |
| 7 | 30,000 | 0.8 | 0.001 | -1 | -0.0712 | 0.5567 | 0.7306 | 2.4334 | 1.3251 | 1.5457 |
| 8 | 300,000 | 0.8 | 0.001 | -1 | 0.0818 | 0.2648 | 0.6272 | 0.6465 | 0.7114 | 0.5339 |
| 9 | 30,000 | 0.3 | 0.010 | 0 | -0.8800 | -0.8934 | -0.9246 | -0.7145 | -0.9552 | -0.8406 |
| 10 | 300,000 | 0.3 | 0.010 | 0 | -0.3909 | -0.0443 | -0.3744 | 2.4960 | 1.0246 | 1.5486 |
| 11 | 30,000 | 0.8 | 0.010 | 0 | -0.8678 | -0.8670 | -0.8660 | -0.0071 | -0.7144 | -0.2569 |
| 12 | 300,000 | 0.8 | 0.010 | 0 | -0.1071 | 0.2484 | -0.1010 | 2.1076 | 1.2397 | 1.5087 |
| 13 | 30,000 | 0.3 | 0.001 | 0 | -0.3302 | 0.6349 | -0.0978 | 3.0676 | 0.8870 | 1.6256 |
| 14 | 300,000 | 0.3 | 0.001 | 0 | 0.0121 | 0.3567 | 0.5443 | 1.1586 | 0.7176 | 0.8452 |
| 15 | 30,000 | 0.8 | 0.001 | 0 | 0.0068 | 0.5925 | 0.5504 | 2.0008 | 1.1088 | 1.5263 |
| 16 | 300,000 | 0.8 | 0.001 | 0 | 0.0372 | 0.2910 | 0.5814 | 0.8824 | 0.7017 | 0.6361 |
| 17 | 30,000 | 0.3 | 0.010 | 1 | -0.9314 | -0.9401 | -0.9535 | -0.7406 | -0.9760 | -0.7875 |
| 18 | 300,000 | 0.3 | 0.010 | 1 | -0.5024 | -0.2669 | -0.4772 | 2.3833 | 0.9816 | 1.6669 |
| 19 | 30,000 | 0.8 | 0.010 | 1 | -0.8261 | -0.9230 | -0.8909 | -0.3114 | -0.7848 | -0.5560 |
| 20 | 300,000 | 0.8 | 0.010 | 1 | -0.1058 | 0.3179 | -0.1140 | 2.1520 | 1.2852 | 1.6884 |
| 21 | 30,000 | 0.3 | 0.001 | 1 | -0.3767 | 0.3615 | -0.2946 | 2.3399 | 1.1233 | 1.7798 |
| 22 | 300,000 | 0.3 | 0.001 | 1 | 0.0689 | 0.3582 | 0.5812 | 1.2500 | 0.6666 | 0.8880 |
| 23 | 30,000 | 0.8 | 0.001 | 1 | 0.0281 | 0.7811 | 0.4752 | 2.1502 | 1.2268 | 1.4006 |
| 24 | 300,000 | 0.8 | 0.001 | 1 | 0.0462 | 0.3228 | 0.5375 | 0.7042 | 0.6861 | 0.4428 |
\(~\) \(~\) \(~\) \(~\)
Relative change in average MSE over all significant SNPs due to method implementation, using a significance threshold of \(5 \times 10^{-8}\) and a skewed distribution of effect sizes:
\(~\) \(~\) \(~\) \(~\)